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Liquidity Premium and Investment Horizons
Equity returns exhibit return predictability tied to trading activity that classical asset-pricing models struggle to explain. A long empirical literature documents that trading volume, order flow, and measures of illiquidity carry information about subsequent stock returns. However, this evidence has developed largely independently of the equilibrium models that explain why trading activity should move prices in the first place. We bring these two strands together: we estimate the price-impact coefficient from Kyle1985 model of strategic informed trading directly from observable equity order flow and test its ability to forecast the cross-section of subsequent stock returns.
In Kyle's model, a single informed trader submits orders against a competitive, risk-neutral market maker who observes only aggregate order flow and prices the asset as a linear function of that flow. The resulting equilibrium price-impact coefficient, $\lambda$, governs how strongly a given unit of net order flow moves prices, and is determined by the ratio of noise-trader variance to fundamental uncertainty about the asset's value. We argue that this equilibrium object, estimated from daily equity trading data, predicts subsequent returns. A stock with a high $\lambda$ is one whose order flow is, in equilibrium, more informative, more costly to trade against, and more sensitive to shifts in investor demand.
We propose a resolution to a long-standing puzzle in this literature: why does trading volume predict returns, and through what mechanism does illiquidity command a premium? Existing empirical work documents a robust association between volume, order flow, and subsequent returns (e.g., LoWang2015, Bajzik2021), but does not, on its own, provide a structural account of why this association should hold or how to translate it into a tractable equilibrium pricing model. Separately, a large literature on equity liquidity premia (AmihudMendelson1986; PastorStambaugh2003; AcharyaPedersen2005) establishes that illiquidity is priced, but typically treats the illiquidity premium as compensation for risk borne by long-horizon investors, leaving open the puzzle first posed by Constantinides1986: if infinite-horizon investors should be indifferent to transaction costs, why is the observed illiquidity premium so large in practice?
We propose that both questions have a common answer rooted in market microstructure theory. Kyle1985's equilibrium implies that $\lambda$ widens precisely when the ratio of informed to noise trading is high relative to fundamental uncertainty, discouraging participation and depressing prices. As order flow normalizes, $\lambda$ narrows and prices recover. This generates a return differential between low- and high-order-flow states that requires no counterparty to knowingly bear a liquidity cost on investors' behalf. The $\lambda$ synamics further offer a resolution to the liquidity premium puzzle grounded in adverse selection rather than risk compensation.
We test four predictions of this framework, each tied to a formal proposition developed in Section 3. We develop signed (directional) order flow as the empirical analog of Kyle's latent order flow variable $y$. First, we posit that signed order flow should positively and significantly predict stock returns since price moves linearly in net order flow in Kyle's equilibrium (Proposition 3). Second, signed order flow should dominate unsigned, aggregate trading volume as a return predictor since $\lambda$ operates on signed flow rather than raw volume. Third, volume volatility, which we use to proxy the noise-trading variance that determines $\lambda$ in equilibrium (Proposition 1), should predict lower subsequent returns since higher noise-trading variance narrows $\lambda$ and degrades the precision of price discovery. Fourth, the predictive content of order flow and $\lambda$ should be strongest at short horizons and high trading frequencies, since the continuous-time extension of Kyle's model implies that price impact rises as the date of full information revelation approaches (Proposition 4).
Using CRSP daily and monthly equity data for the entire CRSP universe from 2020 to 2025, we construct firm-month measures of total trading volume, volume volatility, and signed order flow, along with two estimators of $\hat\lambda_{it}$: a direct within-month regression of price changes on signed order flow and an Amihud2002-style illiquidity ratio. We document four main findings.
First, in panel regressions of monthly stock returns on our volume-based predictors, signed order flow is a strongly significant predictor of both contemporaneous and one-month-ahead returns, robust to the inclusion of standard equity controls (size, book-to-market, momentum, and Amihud illiquidity), while volume volatility predicts returns with a negative sign, consistent with our model's prediction that elevated noise-trading variance narrows $\lambda$ and weakens price discovery.
Second, return-prediction regressions based directly on $\hat\lambda_{it}$ yield informative but specification-sensitive results: the within-month price-impact regression estimator enters significantly and with a large coefficient in our baseline specification with an intercept. However, the sign and magnitude of this relationship vary with specification choices that we investigate as a robustness question in Section 6.
Third, Fama--MacBeth cross-sectional regressions of next-month returns on our predicted-return signal yield a statistically significant average slope, after a Newey--West adjustment, for both our Amihud-style and regression-based $\hat\lambda_{it}$ estimators, indicating that order-flow-based illiquidity measures carry reliable cross-sectional information about future returns over our sample period.
Fourth, we provide a microstructure-based resolution to the liquidity premium puzzle: rather than positing a counterparty who knowingly bears the cost of holding illiquid assets, we show that low signed order flow widens $\lambda$ and depresses prices in equilibrium, while subsequent normalization of order flow narrows $\lambda$ and restores prices, generating the historically observed illiquidity premium as a consequence of equilibrium price impact rather than risk-based compensation.
Our work makes three contributions to asset pricing theory and practice. Methodologically, we provide a direct empirical implementation of Kyle1985's price-impact coefficient using observable daily equity trading data, estimated separately via a regression-based approach and an Amihud-style approach, and show that both carry information for the cross-section of subsequent returns. This bridges the equilibrium market microstructure literature (Kyle1985; GlostenMilgrom1985; EasleyEtAl1996) with the empirical literature on volume and illiquidity in asset pricing (Amihud2002; PastorStambaugh2003). Theoretically, we resolve the liquidity premium puzzle through a demand-based adverse-selection mechanism rather than a risk-compensation mechanism. In doing so, we provide microfoundations for why illiquid assets earn higher returns without requiring implausible assumptions about investor preferences or market segmentation. Empirically, we demonstrate that an equilibrium, microstructure-founded measure of price impact, estimated entirely from CRSP daily price and volume data, has predictive content for monthly stock returns distinct from standard equity characteristics, with direct implications for portfolio construction and market-making.
The paper proceeds as follows. Section 2 reviews the related literature in market microstructure theory, order-flow informativeness, and equity liquidity premia. Section 3 develops our theoretical framework, presenting Kyle1985's equilibrium and four formal propositions on order flow, price impact, and return predictability. Section 4 describes our CRSP-based data construction and summary statistics. Section 5 presents our baseline order-flow and stock-return regressions. Section 6 develops firm-month estimates of Kyle's $\lambda$ and tests their ability to forecast returns. Section 7 compares alternative constructions of $\hat\lambda_{it}$. Section 8 presents an expanding-window out-of-sample forecasting exercise, together with Fama--MacBeth and portfolio-sort robustness checks. Section 9 discusses economic mechanisms and practical implications for portfolio managers and market makers. Section 10 concludes.
Our study touches upon four research streams: (1) market microstructure models of price formation under information asymmetry, (2) empirical measures of order flow and informed trading, (3) the role of trading volume in asset pricing, and (4) liquidity premia in equity markets. We contribute by showing that the equilibrium price-impact coefficient from Kyle1985-style microstructure models, estimated from observable equity order flow, predicts the cross-section of subsequent stock returns and by offering a microstructure-based resolution to the liquidity premium puzzle grounded in adverse selection rather than risk compensation.
Two complementary modeling traditions establish how private information becomes impounded into asset prices through trading. Kyle1985 models a single informed trader submitting market orders against a competitive, risk-neutral market maker who observes only aggregate order flow; in the resulting linear equilibrium, price moves linearly in net order flow with a slope of $\lambda$, the now-canonical measure of price impact. GlostenMilgrom1985 instead models a sequence of individual trades, in which a risk-neutral, zero-expected-profit dealer quotes bid and ask prices that widen purely to compensate for adverse selection from privately informed counterparties. BackBaruch2004 unify the two traditions, showing that Kyle's continuous-time linear equilibrium is the limit of the Glosten--Milgrom sequential-trade equilibrium as trade sizes shrink and arrival rates increase, while Back1992 extends Kyle's framework to continuous time directly, yielding the time-varying price-impact process central to our horizon-dependent predictions (Section 3.5).
A parallel empirical literature develops measures of how much information is embedded in observed trading, independent of the particular equilibrium pricing model. EasleyOHara1987 show that trade size, not just trade direction, carries information: because informed traders prefer to trade larger quantities when their signal is more valuable, market makers rationally widen the price impact of large trades, generating a nonlinear relationship between trade size and price response. EasleyOHara1992 extend the informational content of trading to the time dimension, showing that the absence of trading itself is informative. Intervals without trades signal the absence of new information, so the timing of trade arrivals, not only their size and direction, affects the speed and magnitude of price adjustment.
Building on this framework, EasleyKieferOHaraPaperman1996 develop the probability of informed trading (PIN) measure, a structural estimate of the fraction of order flow attributable to informed traders for a given stock. The estimate is obtained from the sequence of buy and sell trade arrivals. The central empirical finding of EasleyEtAl1996 is that the probability of information-based trading is systematically lower for high-volume stocks, is the direct empirical counterpart of our Proposition 1: higher trading volume is associated with a less favorable informed-to-noise trading ratio $\sigma_u/\sqrt{\Sigma_0}$, and hence a smaller equilibrium $\lambda$. Our use of signed order flow as a proxy for Kyle's latent $y_t = x_t + u_t$ is conceptually continuous with this literature's effort to recover the unobservable informed-trading component from observable trade data.
A separate strand of the microstructure literature asks not how much information trading contains, but where that information is impounded into price when a security trades across multiple venues or related instruments. Hasbrouck1995 develops the information-share methodology, decomposing the innovation in a security's common efficient price into the proportional contributions of each market in which it trades. While our setting is single-market rather than multi-market, the underlying logic is directly relevant: just as Hasbrouck's information share isolates the fraction of price discovery attributable to a given venue's order flow, our $\hat\lambda_{it}$ estimates isolate the fraction of a stock's monthly price variation attributable to signed order flow as opposed to noise-driven trading. This connection motivates a natural robustness check in our empirical design (Section 9): decomposing realized return variance into a signed-order-flow component and a residual component, in the spirit of an information-share calculation applied to a single market over time rather than across markets at a point in time.
The dynamics of trading volume are well documented empirically but remain incompletely understood theoretically. LoWang2015 study volume measures in a portfolio-theoretic context and find that volume does not reduce to a simple measure of institutional flows. LoMamayskyWang2000 examine the informational content of volume from a technical-analysis perspective, finding that price trends accompanied by rising volume are more persistent than price trends unaccompanied by volume. Relevant to our setting, EasleyOHaraSrinivas1998 and PanPoteshman2006 show that option trading volume contains information about subsequent prices of the underlying asset, which PanPoteshman2006 attribute to informed traders exploiting implicit leverage in option markets. We document a related but distinct channel: rather than volume in a derivative market predicting the underlying, we show that signed volume in the underlying equity itself, filtered through a Kyle-style price-impact estimate, predicts the equity's own subsequent returns.
Bajzik2021 synthesizes 44 studies on the volume--return relationship and reports that the impact of volume weakens as stocks mature and their liquidity stabilizes. This finding is consistent with our Proposition 1, in which $\lambda$ depends on the time-varying ratio of informed to noise trading rather than on volume in isolation. BaiEtAl2024 show that volume normalized by amount outstanding helps investors navigate liquidity conditions across related instruments. More broadly, KoijenYogo2019 and CongLiWang2020 motivate pricing financial instruments as a function of their usage rather than purely their fundamentals. We adopt a related premise, treating observed trading activity as informative about price formation in its own right rather than as a secondary characteristic.
The existing literature documents volume--return correlations but does not, on its own, provide a structural account of why volume should predict returns or how to translate that correlation into a tractable pricing model. By grounding our volume measures in Kyle's equilibrium, we supply both the missing theoretical foundation and a direct empirical implementation.
A substantial literature establishes that liquidity is a priced state variable in equity markets. AmihudMendelson1986 show that stocks with wider bid--ask spreads earn higher average returns, a result extended using alternative liquidity proxies including turnover-based measures and the Amihud illiquidity ratio Amihud2002.
AcharyaPedersen2005 propose a liquidity-adjusted capital asset pricing model in which expected returns depend on both an asset's baseline illiquidity and its covariance with aggregate liquidity shocks, while PastorStambaugh2003 show that stocks with high sensitivity to market-wide liquidity deterioration command higher average returns, consistent with a systematic liquidity-risk premium. Johnson2008 distinguishes liquidity (a market's average risk-bearing capacity) from volume (the investor-driven change in that capacity), showing that volume is positively related to the variance of liquidity. This distinction is directly relevant to our use of volume volatility as a proxy for the noise-trading variance $\sigma_u^2$ in Proposition 1.
ReichenbacherSchuster2022 propose size-adapted liquidity measures and show that standard metrics can underestimate illiquidity costs unless trade size is properly normalized, while Cabrol2024 apply machine-learning methods to illiquidity prediction. BrunnermeierPedersen2009 connect funding liquidity (the ability of intermediaries to borrow) to market liquidity, showing how small funding shocks can generate liquidity spirals through forced deleveraging. Such a mechanism is consistent with the regime-dependent strength of our predicted relationships during stress periods (Section 10).
The collective evidence indicates that investors require both a level premium for holding illiquid assets and a separate premium for exposure to systematic liquidity risk. Our contribution is to show that a structural, equilibrium-based measure of illiquidity carries the same pricing information while also nesting a theoretical account of why that information should be priced in the first place.
Our paper also contributes to the literature on the liquidity premium puzzle. Merton1973 hypothesized that investors should demand higher returns to hold illiquid assets to compensate for transaction costs, but he left open which market participants actually bear that cost. Constantinides1986 formalized this as a puzzle: in an equilibrium model, the liquidity premium should be small for investors with an infinite trading horizon, yet large liquidity premia are observed in practice. Chen2022 attributes the discrepancy to heterogeneity in investor information, arguing that better-informed investors are unfazed by transaction costs, while less-informed investors require compensation for illiquidity. FamaFrench1988 separately establish the empirical practice of decomposing security returns into permanent and transitory components, a decomposition we adapt in interpreting the temporary mispricing mechanism below.
Our proposed resolution synthesizes the information-asymmetry mechanism of Kyle1985 and GlostenMilgrom1985 with this puzzle: rather than positing a counterparty who knowingly pays a risk premium to illiquid-asset holders, we show that low signed order flow depresses prices today because Kyle's $\lambda$ widens when noise-trading variance is low relative to informational uncertainty, discouraging participation. Subsequent normalization of order flow and the associated narrowing of $\lambda$ then drives a price recovery, generating a return differential that requires no counterparty to "pay" the premium in equilibrium. This relocates the liquidity premium puzzle's resolution from a risk-compensation story to an adverse-selection and price-impact story, consistent with the market microstructure tradition reviewed above.
In sum, while the existing literature establishes (1) equilibrium models of price formation under information asymmetry, (2) empirical techniques for measuring informed trading and price discovery, and (3) the presence of equity liquidity premia, no prior work directly estimates Kyle's $\lambda$ from observable order flow and tests its ability to forecast the cross-section of equity returns within a unified, microstructure-founded framework. Our contribution fills this gap.
In this section, we propose an agent-driven pricing model. Specifically, we establish that
Our model comprises the following market structure settings:
We assume three types of market participants: Short-horizon traders who rebalance monthly and long-horizon buy-and-hold investors. Among both long- and short-term investors, a fixed proportion $\alpha$ is informed.
We also assume the existence of public and private information. Public information includes disclosed firm fundamentals and debt structure. In addition, some traders are able to observe private firms' signals about near-term liquidity needs. In either case, following EasleyEtAl1996, we assume that the trading volume aggregates all public and private information.
Market structure. A single risky asset has a liquidation value $v \sim N(p_0, \Sigma_0)$. One risk-neutral informed trader observes $v$ perfectly and submits a market order $x$. Noise (liquidity) traders submit an aggregate order $u \sim N(0,\sigma_u^2)$, independent of $v$. A competitive, risk-neutral market maker observes only the total order flow $y = x+u$ (and cannot separate informed from noise flow) and sets the transaction price $p(y)$ to satisfy a zero-expected-profit (semi-strong efficiency) condition.
We restrict attention to linear equilibria of the form
The informed trader chooses $x$ to maximize expected trading profit $\mathbb{E}[(v-p)\,x \mid v]$. The market maker chooses $\lambda$ so that the price equals the conditional expectation of value given observed order flow, $p = \mathbb{E}[v \mid y]$.
We refer to $\lambda$ in (ref) as Kyle's lambda, the standard measure of price impact per unit of (latent) net order flow. In this paper, Kyle's Lambda is an equilibrium object derived from strategic informed trading.
Our direct theoretical implication is that signed/directional volume (Methods 2--3) outperforms simple aggregate volume (Method 1) in predicting short-horizon returns, as predicted by Kyle's model, since $\lambda$ operates on signed order flow $y$, not on unsigned trading volume.
This section provides a dynamic extension of Kyle's model (Kyle1985).
Setup. Trading occurs over $N$ discrete rounds $n=1,\dots,N$ before $v$ is publicly revealed at $T$. At each round, the informed trader submits $\Delta x_n$, noise traders submit $\Delta u_n \sim N(0,\sigma_u^2 \Delta t)$, and the market maker sets $\Delta p_n = \lambda_n \,\Delta y_n$ based on the cumulative order flow to date.
Intuition. A higher $\beta_n$ both (i) directly raises the informed component of flow and (ii) accelerates the market maker's repricing, which, in a sequential-auction equilibrium, raises the rate at which residual uncertainty $\Sigma_n$ is resolved. Faster resolution sustains higher expected order flow in nearby rounds because price still has more "catching up" left relative to $v$.
Propositions (ref)--(ref) imply:
This study uses only the Center for Research in Security Prices (CRSP) daily stock files. Both the dependent variable (stock returns) and the key explanatory variables (order flow, $\hat\lambda$) are constructed entirely from observed equity trading.
For each PERMNO and trading day in the CRSP Daily Stock File (DSF), we obtain the closing price (DlyPrc), share volume (DlyVol), daily return (DlyRet), shares outstanding (ShrOut), and the price and volume adjustment factors (DisFacPr, \texttt{DisFacShr}) needed to construct split-adjusted series. Where available, we also extract daily bid (\texttt{DlyBid}) and ask (\texttt{DlyAsk}) quotes for robustness checks against quoted-spread-based illiquidity measures.
We use the monthly 13-week Treasury bill yield from the Federal Reserve's H.15 Statistical Release as the risk-free rate $r_t$ for the Kyle-lambda regressions and for any excess-return calculations.
We construct the sample as follows:
For each firm $i$ and trading day $\tau$, we define:
These are aggregated to the firm-month level as:
We winsorize all firm-month variables at the 1% and 99% levels to limit the influence of data errors and extreme outliers, consistent with standard practice in the equity microstructure literature.
For the regressions in Sections 6--9 that include standard controls, we additionally construct, using CRSP (and, where indicated, external factor data):
Table (ref) reports the high-level coverage of the filtered universe used in the empirical sections that follow.
Table (ref) reports summary statistics for monthly stock returns, and Table (ref) reports summary statistics for the volume-based variables described above, for the filtered universe.
Table (ref) shows summary statistics for illiquidity measures obtained via 1) Kyle's lambda regression and 2) AMIHUD200231 illiquidity metric.
Figure (ref) summarizes the end-to-end construction process: starting from raw CRSP daily and monthly files, we apply the cleaning and filtering steps in Section 4.2, construct daily order-flow and price-impact variables (Section 4.3), aggregate to the firm-month level, merge in delisting returns and point-in-time index membership, and assemble the final panel used throughout Sections 6--9.
In developing and testing our extended Kyle1985-based equity pricing framework, we rely on several conceptual, methodological, and empirical assumptions. These assumptions guide our theoretical model derivations and inform the design of our empirical tests. Below is a concise summary:
These assumptions underpin our main findings that (1) signed order flow, estimated via Kyle's $\lambda$, captures short-term information asymmetry and price-impact dynamics; (2) daily order flow-driven measures of $\lambda$ significantly improve short-horizon return forecasting relative to aggregate unsigned volume; and (3) the time-varying nature of $\lambda(t)$ implied by the continuous-time extension of Kyle's model retains relevance for explaining longer-horizon predictability through slower-moving aggregate trading-activity measures. Any departure from these assumptions, such as multiple informed traders with heterogeneous signals, time-varying noise-trading variance unrelated to $\sigma_u$, or strategic behavior by the market maker, may alter the empirical outcomes and should be explored in future research.
We estimate two types of regression (contemporaneous and one-month-ahead) for the CRSP equity universe. Let $\text{StockRet}_{it}$ denote the monthly return of stock $i$ in month $t$. We define three order-flow-based explanatory variables, constructed entirely from CRSP daily data:
We estimate:
(A) Contemporaneous Regression:
(B) One-Month-Ahead Regression:
Both regressions are estimated on the full CRSP panel (point-in-time constituents, 2020--2025 or extended where data permit), with and without standard equity controls (log market capitalization, book-to-market, 12-1 month momentum, and the AMIHUD200231 illiquidity ratio), to verify that $\beta_3$ is not subsumed by known characteristics.
\noindentPredicted pattern (Proposition 1 and 3): $\beta_1>0$ but small/insignificant once controls are added; $\beta_2<0$ (noise-trading variance lowers $\lambda$, degrading price discovery); $\beta_3>0$ and robust to controls since signed flow is the direct empirical analog of Kyle's $y$.
We now construct firm-month estimates of Kyle's $\lambda$ directly.
For each firm $i$ and month $t$, we estimate $\hat\lambda_{it}$ via an intramonth price-impact regression:
where $\text{OF}_{i\tau} = \text{Volume}_{i\tau} \times \operatorname{sign}(\Delta P_{i\tau})$ is the daily signed order-flow proxy and $\hat\lambda_{it}$ is the estimated slope, taken as the firm-month Kyle-lambda estimate. As a robustness check, we also compute the Amihud-style ratio
which proxies price impact without requiring within-month regression and is directly comparable to the existing illiquidity literature (AMIHUD200231).
We estimate, with and without an intercept:
Having established that $\hat\lambda_{it}$ predicts returns, we compare two constructions.
Method A (Amihud-style, level):
Method B (Kyle price-impact regression, signed):
Each $\hat\lambda^{Method}_{it}$ is used as the sole regressor in Equation (ref), estimated with and without an intercept.
\noindentPredicted pattern: We expect Method B, which incorporates signed price-volume information, to significantly outperform Method A.
We adopt the same expanding (rolling) window design, with the predictor variable replaced.
For each firm, we initialize a training window covering the first 30% of chronological observations. At each step:
We pool all out-of-sample predicted and actual returns across firms and estimate the final regression
using only out-of-sample observations, separately for each $\lambda$-construction method (A and B) defined in Section 8, and separately for a High-Frequency estimation window (daily intramonth regression, Method B) versus a Low-Frequency estimation window (monthly aggregate, Method A). This test is motivated by Proposition 4's prediction that price impact and its predictive content are strongest at short horizons/high frequency.
Pooled OLS $R^2$ for monthly equity returns is expected to be small (consistent with FamaFrench1988,chen2007, and the broader asset-pricing literature on monthly return predictability). We therefore complement the pooled regression with:
Fama--MacBeth cross-sectional regressions. For each month $t$:
summarized by the time-series mean of $\beta_t$ and its Newey--West-adjusted $t$-statistic.
Decile portfolio sorts. Each month, sort stocks into deciles on the model-implied $\hat\lambda_{it}$ signal from the expanding-window procedure; report long-only (D10) and long--short (D10--D1) monthly Sharpe ratios, separately for the High-Frequency and Low-Frequency legs.
Based on our analysis, volume affects returns along 4 distinct mechanisms:
Our analysis provides a microstructure-based resolution to the liquidity premium puzzle that requires neither risk-based compensation nor market segmentation. We review the puzzle and then show how our framework resolves it.
\paragraph{The puzzle and its classical formulations.} Amihud and Mendelson (1986) show empirically that stocks with wider bid-ask spreads earn higher average returns and argue that the premium compensates investors for the transaction costs they incur. Merton (1973) provides a theoretical foundation: investors require higher expected returns on illiquid assets precisely because they bear the cost of trading those assets. The puzzle, formalized by Constantinides1986, is that this argument breaks down in general equilibrium: an investor with an infinite trading horizon should be nearly indifferent to transaction costs, yet large illiquidity premia persist in the data. As Constantinides1986 shows, the liquidity premium that investors actually require in equilibrium should be negligibly small, far below what empirical estimates suggest. Recent work by Chen2022 attributes the gap to heterogeneity in investor information: better-informed investors are indifferent to transaction costs while less-informed investors require compensation. However, this leaves open the question of why uninformed investors should hold illiquid assets in the first place if those assets expose them to greater adverse-selection losses.
\paragraph{Our resolution: adverse selection and temporary mispricing.} We propose that the illiquidity premium does not need to be required by any investor in equilibrium; instead, it is realized as a consequence of temporary mispricing driven by the equilibrium widening and subsequent narrowing of Kyle's $\lambda$.
The mechanism has two stages, tied directly to Propositions (ref) and (ref):
This mechanism resolves the Constantinides (1986) paradox directly: the illiquidity premium is realized, not required. No investor needs to demand a risk premium for holding an illiquid asset; they simply purchase it when adverse selection drives prices below fundamental value and hold it until the order-flow process restores prices. No counterparty knowingly pays the premium. Instead, the premium arises from the equilibrium dynamics of $\lambda$ itself, which depresses and then restores prices mechanically as order flow evolves. This also answers Chen2022 open question about why uninformed investors hold illiquid assets: they do not need to hold them during the low-order-flow stage; the return accrues to those who are willing to provide liquidity precisely when $\lambda$ is widest, in the manner of a contrarian market maker rather than a passive buy-and-hold investor.
\paragraph{Empirical consistency.} This mechanism is consistent with four features of our empirical results. First, the negative and significant coefficient on signed order flow in our one-month-ahead regressions (Table 5) reflects the Stage 1 mechanism: lower signed flow today predicts lower returns tomorrow, as $\lambda$ widens and prices continue to adjust downward before recovering. Second, the negative coefficient on volume volatility, our proxy for $\sigma_u^2$ in Proposition (ref), is consistent with the prediction that elevated noise-trading variance narrows $\lambda$, attenuating the adverse-selection-driven discount and, therefore, reducing the magnitude of the subsequent price recovery. Third, the Fama--MacBeth slopes documented in Table 10 are negative on average across both $\hat\lambda$ estimators after Newey--West adjustment. This reflects the cross-sectional manifestation of this mechanism: stocks with high $\hat\lambda_{it}$ today (widened price impact, depressed prices) earn lower subsequent returns over our sample period, consistent with the transition from Stage 1 to Stage 2 having already occurred in the month following the wide-$\lambda$ episode. Fourth, our resolution does not require market segmentation, implausible preferences, or clientele effects: it follows directly from the equilibrium structure of Kyle1985's model applied to a single, unified equity market.
\paragraph{Distinction from risk-based explanations.} Our resolution differs from the risk-based approach in a subtle but important respect. Standard theories attribute the illiquidity premium to required compensation: investors know ex ante that illiquid assets carry higher transaction costs or greater exposure to systematic liquidity risk AcharyaPedersen2005,PastorStambaugh2003, and they demand a higher expected return before holding those assets. Our theory attributes the premium to realized returns from temporary mispricing: investors do not need to anticipate a premium; they simply find that assets which experienced low signed order flow subsequently recover in price as $\lambda$ normalizes. The premium is a consequence of equilibrium price-impact dynamics, not of risk aversion or market frictions per se. This distinction has a testable implication: if the risk-based story is correct, the illiquidity premium should be predictable ex ante and should persist unconditionally. If our adverse-selection story is correct, the premium should be concentrated in periods following low signed order flow and should attenuate once $\lambda$ has normalized.
We develop an equity asset-pricing framework grounded in Kyle's (1985) model of strategic informed trading, in which the equilibrium price-impact coefficient $\lambda$ governs how trading activity is impounded into prices. Using CRSP daily and monthly data for all equities in the CRSP universe from 2020 to 2025, we construct firm-month estimates of signed order flow and of $\lambda$ itself, via both a direct within-month price-impact regression and an Amihud-style illiquidity ratio. We then test their ability to forecast the cross-section of subsequent stock returns.
Three findings emerge.
First, signed order flow is a strong and highly significant predictor of both contemporaneous and one-month-ahead stock returns, robust to the inclusion of standard equity controls (size, book-to-market, momentum, and Amihud illiquidity), consistent with Proposition 3's prediction that order-flow innovations move prices in equilibrium.
Second, volume volatility, our proxy for the noise-trading variance $\sigma_u^2$ in Proposition 1, predicts lower subsequent returns, consistent with the model's prediction that a higher noise-to-information trading ratio widens $\lambda$ and degrades price discovery.
Third, our return-prediction regressions based on $\hat\lambda_{it}$ itself are informative but specification-sensitive. results vary with the inclusion of an intercept and across our two $\lambda$ estimators, a pattern we investigate but do not yet fully resolve, and which we flag explicitly as a target for the robustness work in the sections that follow. Fama--MacBeth cross-sectional regressions of next-month returns on the predicted-return signal yield a statistically significant average slope (Newey--West $t$-statistics of $-2.27$ and $-2.25$ for the Amihud-style and regression-based estimators, respectively, over 58 months), indicating that order-flow-based illiquidity measures carry reliable, if modest, cross-sectional information about future returns over our sample.
Theoretically, our framework offers a resolution to the liquidity premium puzzle of Constantinides1986 that relocates the source of the illiquidity premium from risk-based compensation to an adverse-selection and price-impact mechanism. Low signed order flow widens $\lambda$ and depresses prices today, as Kyle's market maker rationally demands greater compensation for the elevated risk of trading against informed counterparties; subsequent normalization of order flow narrows $\lambda$ and restores prices, generating the return differential historically attributed to a liquidity premium without requiring any counterparty to knowingly bear that cost in equilibrium. This relocates the puzzle's resolution from the risk-compensation story of amihud86 to the market microstructure tradition of Kyle1985 and GlostenMilgrom1985.
For practitioners, our results suggest that equity portfolio managers running short-horizon or high-turnover strategies should favor stocks with low $\hat\lambda_{it}$ to minimize expected trading costs, while market makers can use real-time estimates of signed order flow to recalibrate quoted spreads in response to changes in the informed-to-noise trading ratio that determines $\lambda$ in equilibrium. More broadly, our findings indicate that high-frequency, order-flow-based measures of price impact, estimated entirely from publicly available equity trading data, provide a theoretically motivated and empirically tractable complement to existing illiquidity proxies such as the Amihud ratio and the bid-ask spread.